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#### karush

##### Well-known member

- Jan 31, 2012

- 3,068

$\tiny{311.1.3.12}$

Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$

$ a_1\left[\begin{array}{r} 1\\0\\1 \end{array}\right],

a_2\left[\begin{array}{r} -2\\3\\-2 \end{array}\right],

a_3\left[\begin{array}{r} -6\\7\\5 \end{array}\right],

b=\left[\begin{array}{r} -7\\13\\4 \end{array}\right]$

ok I don't think this is too difficult to do.

but these matrix problems are very error prone

so thot I would just do a step at a time here

from the example I looked at this is the same thing as

$\left[\begin{array}{lll}a_1&+(-2a_2)&+(-6a_3)\\

&+3a_2 &+7a_3\\

a_1&+(-2a_2)&+5a_3) \end{array}\right]

=\left[\begin{array}{r} -7\\13\\4 \end{array}\right]$

I left all the + signs in since I think this is what a combination is, so then

$\left[\begin{array}{rrr|r}1&-2&-6&-7\\

0&3&7&13\\

1&-2&5&4 \end{array}\right]$

by RREF I got $a_1=3,\quad a_2=2\quad a_3=1$

Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$

$ a_1\left[\begin{array}{r} 1\\0\\1 \end{array}\right],

a_2\left[\begin{array}{r} -2\\3\\-2 \end{array}\right],

a_3\left[\begin{array}{r} -6\\7\\5 \end{array}\right],

b=\left[\begin{array}{r} -7\\13\\4 \end{array}\right]$

ok I don't think this is too difficult to do.

but these matrix problems are very error prone

so thot I would just do a step at a time here

from the example I looked at this is the same thing as

&+3a_2 &+7a_3\\

a_1&+(-2a_2)&+5a_3) \end{array}\right]

=\left[\begin{array}{r} -7\\13\\4 \end{array}\right]$

I left all the + signs in since I think this is what a combination is, so then

$\left[\begin{array}{rrr|r}1&-2&-6&-7\\

0&3&7&13\\

1&-2&5&4 \end{array}\right]$

by RREF I got $a_1=3,\quad a_2=2\quad a_3=1$

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